PUBLICATIONS MATHÉMATIQUES DE L'IHES

Online first
MLC at Feigenbaum points
Dzmitry Dudko1 ; Mikhail Lyubich1
1Institute for Mathematical Sciences, Mathematics Department Stony Brook University, 11733 Stony Brook, USA
Publications Mathématiques de l'IHÉS, Online first, pp. 97-141
PDF

We prove a priori bounds for Feigenbaum quadratic polynomials, i.e., infinitely renormalizable polynomials $f_c: z\mapsto z^2+c$ of bounded type. It implies local connectivity of the corresponding Julia sets $J(f_c)$ and MLC (local connectivity of the Mandelbrot set $\mathcal{M}$) at the corresponding parameters $c$. It also yields the scaling Universality, dynamical and parameter, for the corresponding combinatorics. The MLC Conjecture was open for the most classical period-doubling Feigenbaum parameter as well as for the complex tripling renormalizations. Universality for the latter was conjectured by Goldberg–Khanin–Sinai in the early 1980s.

Received:
Revised:
Accepted:
Online First:
DOI: 10.5802/pmihes.2

Dzmitry Dudko  1 ; Mikhail Lyubich  1

1 Institute for Mathematical Sciences, Mathematics Department Stony Brook University, 11733 Stony Brook, USA 
@unpublished{10_5802_pmihes_2,
     author = {Dzmitry Dudko and Mikhail Lyubich},
     title = {MLC at {Feigenbaum} points},
     journal = {Publications Math\'ematiques de l'IH\'ES},
     year = {2026},
     publisher = {IHES},
     doi = {10.5802/pmihes.2},
     language = {en},
     note = {Online first},
}
TY  - UNPB
AU  - Dzmitry Dudko
AU  - Mikhail Lyubich
TI  - MLC at Feigenbaum points
JO  - Publications Mathématiques de l'IHÉS
PY  - 2026
PB  - IHES
N1  - Online first
DO  - 10.5802/pmihes.2
LA  - en
ID  - 10_5802_pmihes_2
ER  - 
%0 Unpublished Work
%A Dzmitry Dudko
%A Mikhail Lyubich
%T MLC at Feigenbaum points
%J Publications Mathématiques de l'IHÉS
%D 2026
%V 0
%I IHES
%Z Online first
%R 10.5802/pmihes.2
%G en
%F 10_5802_pmihes_2
Dzmitry Dudko; Mikhail Lyubich. MLC at Feigenbaum points. Publications Mathématiques de l'IHÉS, Online first, pp. 97-141

[1] Lars V. Ahlfors Conformal Invariants: Topics in Geometric Function Theory, McGraw-Hill Series in Higher Mathematics, McGraw-Hill, 1973 | MR | Zbl

[2] Artur Avila; Mikhail Lyubich The full renormalization horseshoe for unimodal maps of higher degree: exponential contraction along hybrid classes, Publ. Math., Inst. Hautes Étud. Sci., Volume 114 (2011), pp. 171-223 | DOI | Zbl | Numdam | MR

[3] Laurent Bartholdi; Dzmitry Dudko Algorithmic aspects of branched coverings I/V. Van Kampen’s Theorem for bisets, Groups Geom. Dyn., Volume 12 (2018) no. 1, pp. 121-172 | DOI | Zbl | MR

[4] Pierre Coullet; Charles Tresser Itérations d’endomorphismes et groupe de renormalisation, Comptes Rendus. Mathématique, Volume 287 (1978), pp. 577-580 | Zbl

[5] Pierre Coullet; Charles Tresser Itérations d’endomorphismes et groupe de renormalisation, J. Phys. Colloques, Volume 39 (1978) no. C5, pp. 25-28 | DOI

[6] Adrien Douady; John Hamal Hubbard Étude dynamique des polynômes complexes (première partie), Publications Mathématiques d’Orsay, 84-02, Université de Paris-Sud, 1984, 80 pages | Zbl | MR

[7] Adrien Douady; John Hamal Hubbard Étude dynamique des polynômes complexes (deuxième partie), Publications Mathématiques d’Orsay, 85-04, Université de Paris-Sud, 1985, 162 pages | Zbl

[8] Adrien Douady; John Hamal Hubbard On the dynamics of polynomial-like maps, Ann. Sci. Éc. Norm. Supér. (4), Volume 18 (1985), pp. 287-343 | DOI | Zbl | MR

[9] Dzmitry Dudko; Mikhail Lyubich Uniform a priori bounds for neutral renormalization (2022) (https://arxiv.org/abs/2210.09280v1) | Zbl

[10] Dzmitry Dudko; Mikhail Lyubich Local connectivity of the Mandelbrot set at some satellite parameter of bounded type, Geom. Funct. Anal., Volume 33 (2023) no. 4, pp. 912-1047 | DOI | Zbl | MR

[11] Dzmitry Dudko; Mikhail Lyubich; Nikita Selinger Pacman renormalization and self-similarity of the Mandelbrot set near Siegel parameters, J. Am. Math. Soc., Volume 33 (2020) no. 3, pp. 653-733 | DOI | Zbl | MR

[12] Benson Farb; Dan Margalit A primer on mapping class groups, Princeton Mathematical Series, 49, Princeton University Press, 2012 | MR | Zbl

[13] Mitchell J. Feigenbaum Quantitative universality for a class of nonlinear transformations, J. Stat. Phys., Volume 19 (1978), pp. 25-52 | DOI | Zbl | MR

[14] Mitchell J. Feigenbaum The universal metric properties of nonlinear transformations, J. Stat. Phys., Volume 21 (1979), pp. 669-706 | DOI | Zbl | MR

[15] Andrey I. Gol’dberg; Konstantin M. Khanin; Yakov G. Sinai Universal properties of sequences of period-tripling bifurcations, Usp. Mat. Nauk, Volume 38 (1983) no. 1, pp. 159-160 | MR | Zbl

[16] John H. Hubbard Local connectivity of Julia sets and bifurcation loci: three theorems of J.-C. Yoccoz, Topological Methods in Modern Mathematics, A Symposium in Honor of John Milnor’s 60th Birthday, Publish or Perish Inc., 1993, pp. 467-511 | MR | Zbl

[17] Yutaka Ishii; John Smillie Homotopy shadowing, Am. J. Math., Volume 2010 (2010) no. 4, pp. 987-1029 | DOI | Zbl | MR

[18] Jeremy Kahn A priori bounds for some infinitely renormalizable quadratics: I. Bounded primitive combinatorics (2006) (Preprint Stony Brook, # 5) | arXiv | Zbl

[19] Jeremy Kahn; Micha Lyubich A priori bounds for some infinitely renormalizable quadratic: IV. Elephant Eyes (2026) | arXiv | Zbl

[20] Jeremy Kahn; Mikhail Lyubich A priori bounds for some infinitely renormalizable quadratics: II. Decorations, Ann. Sci. Éc. Norm. Supér. (4), Volume 41 (2008) no. 1, pp. 57-84 | DOI | Zbl | Numdam | MR

[21] Jeremy Kahn; Mikhail Lyubich A priori bounds for some infinitely renormalizable quaratics: III. Molecules, Complex Dynamics: Families and Friends. Proceeding of the conference dedicated to Hubbard’s 60th birthday (Dierk Schleicher, ed.), A K Peters (2009), pp. 229-254 | DOI | MR | Zbl

[22] Jeremy Kahn; Mikhail Lyubich Quasi-Additivity Law in Conformal Geometry, Ann. Math. (2), Volume 169 (2009) no. 2, pp. 561-593 | DOI | Zbl | MR

[23] Mikhail Lyubich Conformal Geometry and Dynamics of Quadratic Polynomials, vol I-II (Book in preparation; www.math.stonybrook.edu/~mlyubich/book.pdf)

[24] Mikhail Lyubich Dynamics of quadratic polynomials, I-II, Acta Math., Volume 178 (1997) no. 2, pp. 185-297 | DOI | Zbl | MR

[25] Mikhail Lyubich Feigenbaum–Coullet–Tresser Universality and Milnor’s Hairiness Conjecture, Ann. Math. (2), Volume 149 (1999) no. 2, pp. 319-420 | DOI | Zbl

[26] Mikhail Lyubich Story of the Feigenbaum point, Talk at the CIRM conference “Advancing Bridges in Complex Dynamics”, https://library.cirm-math.fr/Record.htm?idlist=1&record=19289042124910072249, 2021

[27] Curtis T. McMullen Complex dynamics and renormalization, Annals of Mathematics Studies, 135, Princeton University Press, 1995 | DOI | Zbl | MR

[28] Curtis T. McMullen Renormalization and 3-manifolds which fiber over the circle, Annals of Mathematics Studies, 142, Princeton University Press, 1996 | DOI | Zbl | MR

[29] John Milnor Local connectivity of Julia sets: expository lectures, The Mandelbrot Set, Themes and Variations (Tan Lei, ed.) (London Mathematical Society Lecture Note Series), Volume 274, Cambridge University Press, 2000, pp. 67-116 | Zbl | DOI

[30] Volodymyr Nekrashevych Combinatorial models of expanding dynamical systems, Ergodic Theory Dyn. Syst., Volume 34 (2014) no. 3, pp. 938-985 | DOI | Zbl | MR

[31] Dennis Sullivan Quasiconformal homeomorphisms in dynamics, topology and geometry, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986), American Mathematical Society, 1987, pp. 1216-1228 | MR | Zbl

[32] Dennis Sullivan Bounds, quadratic differentials and renormalization conjectures, Mathematics into the twenty-first century. Proceedings of the AMS centennial symposium, Providence, RI, USA, August 8–12, 1988 (Felix E. Browder, ed.) (American Mathematical Society Centennial Publications), Volume 2, American Mathematical Society, 1992, pp. 417-466 | Zbl

[33] Dylan P. Thurston A positive characterization of rational maps, Ann. Math. (2), Volume 192 (2020) no. 1, pp. 1-46 | Zbl | DOI

Cited by Sources: